Human Eye and Colourful World — Practice

Question 1

What is the least distance of distinct vision, and how does it change with age?

Accepted Answer

It is the closest distance at which an object can be seen comfortably and distinctly, about 25 cm for a healthy adult. Below about 10 years the eye muscles are strong and flexible, so it can be as close as 7 to 8 cm. In old age the muscles cannot sustain strain, so it shifts to about 1 to 2 m or even more.

Question 2

A doctor advises the use of a 2D lens. What is its focal length?

Accepted Answer

Using P = 100/f with f in centimetres, 2 = 100/f, so f = 50 cm.

Question 3

Why does the pupil of the eye appear black?

Accepted Answer

Any light falling on the pupil goes into the eye, and there is almost no chance of it coming back out. With essentially no light returning from it to the observer, the pupil appears black.

Question 4

Show that the maximum and minimum focal lengths of the eye lens are 2.5 cm and 2.27 cm.

Accepted Answer

The image distance is fixed at v = 2.5 cm. For an object at infinity, u = -infinity, so 1/f = 1/2.5 + 0 and f max = 2.5 cm. For an object at the least distance of distinct vision, u = -25 cm, so 1/f = 1/2.5 + 1/25 = 11/25, giving f min = 25/11 = 2.27 cm. The eye lens therefore works between 2.27 cm and 2.5 cm.

Question 5

Explain accommodation of the eye lens.

Accepted Answer

Accommodation is the ability of the eye lens to change its focal length. The ciliary muscles attached to the lens change its radii of curvature. For a distant object the muscles relax and the focal length takes its maximum value, equal to the lens-to-retina distance. For a nearer object the muscles strain and the focal length decreases, so that the image still forms on the retina. The muscles cannot strain beyond a limit, which is why there is a least distance of distinct vision.

Question 6

A prism with an angle A of 60 degrees produces an angle of minimum deviation of 30 degrees. Find the refractive index of its material.

Accepted Answer

Using n = sin[(A + D)/2] / sin(A/2) with A = 60 and D = 30, the numerator is sin(90/2) = sin 45 = 1/root2 and the denominator is sin(60/2) = sin 30 = 1/2. So n = (1/root2) divided by (1/2) = 2/root2 = root2. The refractive index is root 2, about 1.414.

Question 7

Why does the angle of deviation first decrease and then increase as the angle of incidence is increased?

Accepted Answer

The deviation at each surface depends on the angle of incidence there, and the two contributions change in opposite senses as the ray is rotated. Starting from a small angle of incidence, the total deviation falls until the ray inside the prism runs parallel to the base, where i1 = i2 and r1 = r2. Beyond that point the deviation rises again. The lowest point of the curve is the angle of minimum deviation D, found on the graph by drawing a tangent parallel to the x-axis.

Question 8

Why can dispersion not be explained by ray theory, and what explanation does wave theory give?

Accepted Answer

Ray theory assigns a prism a single refractive index, so there can be only one angle of minimum deviation, and by Fermat's principle light takes the single path of least time. Yet white light through a prism visibly takes several different paths at once, which ray theory cannot accommodate. Wave theory treats white light as a collection of waves of different wavelengths, violet shortest and red longest. The speed of light in vacuum is the same for all colours, but in a medium it depends on wavelength, and since refractive index is the ratio of these speeds, refractive index depends on wavelength. Each colour then follows its own least-time path and refracts by a different amount, so the colours separate. Experimentally refractive index decreases as wavelength increases, so red deviates least and violet most.

Question 9

Explain why the sun appears red at sunrise and sunset but white at noon.

Accepted Answer

Scattering by a particle is effective when the particle size is comparable to the wavelength of the light. Molecules of a size comparable to the wavelength of red light are few in the atmosphere, so red is scattered the least of all colours. At sunrise and sunset the sunlight must travel a much longer path through the atmosphere to reach the observer, and along that path every colour except red is scattered away before arriving. Only the weakly scattered red survives, so the sun looks red. At noon the path through the atmosphere is much shorter, so all colours arrive without much scattering and combine, and the sun appears white.

Question 10

Why is a rainbow seen as a bow, and why does each raindrop contribute only one colour?

Accepted Answer

Inside a drop, sunlight refracts on entry and disperses, reflects by total internal reflection at the far side, and refracts again on leaving, which widens the separation between red and violet. The angle between the incoming and outgoing rays reaches a maximum of about 42 degrees, and the rainbow is bright near that angle. Red emerges at 42 degrees and violet at 40 degrees. An observer therefore sees red only from drops lying at 42 degrees to the sunbeam and violet only from drops at 40 degrees, so any single drop can send only one colour to the eye; the red from a violet-sending drop passes below the eye. The set of drops at a given angle forms a cone with its tip at the observer's eye, with red on the outermost layer and violet innermost, and it is the visible part of that cone that appears as a bow.

Practice — Human Eye and Colourful World

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